So - is Theravada Buddhism the only modern representative on Hinayana Buddhism or are there another current branches of Buddhism, no matter how minor, that could also be termed 'Hinayana'. Or in other words would Theravada be completely synonymous with 'currently existing Hinayana Buddhism'.

4 Answers
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You are comparing apples and oranges. Theravada is not Hinayana. Some of Theravada practitioners are at Mahayana level, and some Tibetan buddhists are at Hinayana level.

When one's understanding of (and attitude towards) Buddha-Dharma is very simplistic and literal, we say such person is a Hinayana-level practitioner. On Hinayana level we get indoctrinated with basic concepts like Samsara and Nirvana, learn to follow basic discipline, like Five Precepts and overall minimalism, and establish basic meditation skills, like Samatha meditation on breathing. At this level everything is cut and dry, all instructions are well defined, there is no room for hesitation or sloppiness anywhere. One's focus is on one's practice, and on overcoming one's pathological habits, esp. the habit to give in to random impulses.

When one has established the foundation, one can either get stuck on this precise well-defined formulaic black-and-white level, and remain a (childish) Hinayana ascetic forever, or go onto more advanced aspects of Buddhism -- more subtle and fuzzy when you speak about them, but much more direct in terms of actual experience. Again I emphasize, this happens even in Theravada! At this level, one's focus shifts from self to others, and from overcoming impulses to overcoming aversions. At the same time, as one keeps on applying Dharma to one's daily life, one's realization slowly matures to the point when one no longer needs formulaic instructions and can handle issues with a sense of wisdom. This is so-called Mahayana attitude to Buddha-Dharma.

Hinayana is complicated philosophical term, sometimes it refer to lower motivation, sometimes to some misunderstanding of Teaching, sometimes to smaller knowledge, or restricted acceptance of Buddha's Teaching, and sometimes it's misnomer for early schools. I want to stress, that term is philosophical from just a few of Mahayana sutras. It is not a simple thing as school, (and Mahayana is not a school too).

Hinayana as type of motivation (lower than reaching Buddhahood) is acceptable lower motivation in Mahayana schools for some types of personalities. So, in that sense Hinayana is survived inside of any Mahayana school, which accepts such notion.

In the sense of non-Mahayana: many Mahayana schools accept and their canons contain non-Mahayana teachings. All traditional Mahayana schools actually have survived vinaya lineages of early schools, not related to Mahayana doctrines.

On the other hand, Theravada developed bodhisatta practices and philosophical concepts which thought by some to be distinctively mahayanistic. It just don't generally accepts Mahayana canon, but don't officially denies either. So it's personal choice to accept it or not. Some Theravada scholars studied Mahayana, for example famous Walpola Rahula, he even wrote dissertation on Abhidharmasmuccaya with parallels to Pali canon.

Theravada Buddhism is not a representative of Hinayana. Just because some choose to call a derogatory term to certain traditions, doesn't make them representatives of the term. ex: If someone calls you a peasant, does it make you a representative of the word? Even if one does not mean it as an insult anymore, it's technically incorrect to use the word as Theravada Buddhism clearly recognizes and supports attaining enlightenment through all three Bodhis.

Bhikku Bodhi, an ordained monk of the Theravada tradition wrote an essay titled "Arahants, Bodhisattvas, and Buddhas". This essay describes his view on the relationship between these three and how the bodhisattva ideal emerged from the early Buddhist sects and how they are viewed within his tradition.

In summary, based on the earliest records of the the Buddha's teachings, he never discussed the bodhisattva path or the arahant path. But through considering the amount of karma that would be necessary to become someone as exalted as the Buddha, the bodhisattva path was envisioned. Although this path was never discussed in the early Buddhist scriptures, it was accepted as a logical extension of the teachings.

During the age of Sectarian Buddhism, the Early Buddhist schools came to admit three "vehicles" to enlightenment: the vehicle of the disciple arahant, the śrāvaka-yāna, to be taken by the greatest number of disciples; the vehicle of the "solitary enlightened one" who attains realization without a teacher but does not teach, the pratyekabuddha-yāna, which is still more difficult; and the vehicle of the aspirant to Buddhahood, the bodhisattva-yāna. Once it became widespread in mainstream Indian Buddhism, the idea of the three vehicles was not only taken up by the Mahāyāna but was eventually also absorbed into conservative Theravāda Buddhism.

However, although the bodhisattva-yana is recognized in the Theravada tradition, the sravaka-yana is still the path that they advocate for most in accordance with the Buddha's advice. In this sense, it could be called Hinayana, but when you talk about a "lesser" path it's important to realize what it is less than and what it is greater than.

The bodhisattva-yana is truly a difficult path. It's like running into a burning house over and over again to save as many people as can be saved even though you will be burned. Or guarding the mouth of a cave full of demons day and night so that you can warn people away from entering, even though you will get bitten. It's not something that most people are capable of.

The sravaka-yana is far greater than the path followed by an ordinary person, a foolish person, or an animal. It's like being in a burning house and helping as many people as possible to safety, before saving yourself as well. It's like guarding the mouth of a cave full of demons during the day to warn as many people as possible, but retreating to safety at night.

@ChrisW Thanks for the information. The way he describes the Mahayana does disagree with my (admittedly immature) understanding. I'll have to do some research and think about updating my answer.
– Nolan LuckettFeb 4 '19 at 20:07

I'm not saying your answer is wrong at all, only that he is maybe not the most reliable explainer of Mahayana. I guess it's an accurate explanation of the Theravada view of what the bodhisattva-yana is.
– ChrisW♦Feb 4 '19 at 20:12